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Gauge Fixing and Renormalization in Topological Quantum Field Theory*

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Gauge Fixing and Renormalization in Topological Quantum Field Theory* SLAC-PUB-4630 May 1988 P3 Gauge Fixing and Renormalization in Topological Quantum Field Theory* ROGER BROOKS, DAVID MONTANO, AND JACOB SONNENSCHEIN+ Stanford Linear Accelerator Center Stanford University, Stanford, California 94 309 ABSTRACT In this letter we show how Witten’s topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero. Submitted to Phys. Lett. * Work supported by the Department of Energy, contract DE-AC03-76SF00515. + Work supported by Dr. Chaim Weismann Postdoctoral Fellowship. 1. INTRODUCTION Recently, there has been an effort to study the topology of four dimensional manifolds using the methods of quantum field theory!‘-‘] This was motivated by Donaldson’s use of self-dual Yang-Mills equations to investigate the topology of low dimensional manifolds!” The work of Floer on three manifolds, as interpreted by Atiyah!“can be understood as a modified non-relativistic field theory. Atiyah then conjectured that a relativistic quantum field theory could be used to study Donaldson’s invariants on four manifolds. This led Witten to work out a series of preprints in which just such topological quantum field theories (TQFT) were described.14-61 In the first paper a TQFT was proposed whose correlation func- tions were purely topological and reproduced Donaldson’s invariants!Q’ A later paper extended these ideas to a more complicated Lagrangian which could be used to study similar invariants in topological gravity!51These TQFT’s all pos- sess a fermionic symmetry analogous to a BRST symmetry. In this letter we will show how both of these theories may be obtained through a very simple gauge fixing procedure. A prescription for how quantum field theory calculations may be done for the TQFT will be presented. These calculations will allow us to study the perturbative renormalizability of the theory. It will be shown that the beta function for the coupling constant is not zero. This suggests that there is a quantum contribution to the stress-energy tensor which may indicate a metric dependence of the partition function. Throughout this work we will use the notations of refs. [4] and [5]. However, in our discussion of the Yang-Mills TQFT, actions will be defined in terms of Lagrangians via I = s d4x ,/$3-C, where g is the determinant of the metric. 2. WITTEN’S TOPOLOGICAL YANG-MILLS THEORY We begin by deriving Witten’s Lagrangian for the non-abelian Yang-Mills (YM) theory’*‘by gauge fixing the local transformation &A,” = 8,” . (2.1) Here, Aaa is a gauge field in the adjoint representation (index a). The gauge fixing will be done using the following principles as guidelines: First, the BRST gauge fixing procedure will be employed in order that we manifestly maintain the fermionic symmetries of ref. [4]. Second, because of their importance in the latter work, we will seek to preserve the scaling and U symmetries. Finally, we would like a YM invariant action at the end of the process. *Explicit coupling constant (ee) dependence will be given in our expressions. For this we define the YM covariant derivative as D, G d, + eo[Aa, 1. We have found that the gauge fixing ansatz which leads to Witten’s theory is Fap + tap E 0, where k,p = &QI+F~~ is the dual of the YM field strength, F Q. To see this, let us assume that there is some Lagrangian, &, which is both YM and 61 invariant. Two examples are fZo = 0 and fZo oc [F,pbfl]. Both of these are topological with the Euclidean integral of the second yielding the Pontryagin index (winding number). The former is trivially invariant while the latter is invariant under ;I transformations if: (i) the Bianchi Identity for the YM field-strength, D,i@ = 0, is used, and (ii) the parameter 8, asymptotically drops off as one power faster than the gauge field. This is the requirement that 8, does not change the Pontryagin number. Given, lo, we BRST gauge fix by writing down the gauge fixing and Faddeev-Popov Lagrangians: where we have introduced the BRST operator, iI, through the definition 61 z 3 i&l, for some constant anti-commuting parameter, E. Traces in the adjoint rep- resentation are implied in all of our Lagrangians. This particular Lagrangian maintains the BRST symmetry: A 61&z" =hxa , &baa = 0 , (2.3) iIx aPa -B= ‘-@“, &B al3a -0- . The anti-ghost field, x”p, and the BRST auxiliary field, Bap are anti-symmetric and self-dual. The field, $a, is the ghost conjugate to x@. A few remarks about eqns. (2.2) and (2.3) are in order. To obtain Witten’s theory, we must pick the gauge oe = 0 for the gauge parameter. The operator 81 yields the usual off-shell, nilpotent, BRST algebra. The equation of motion of this auxiliary field yields the transformation law for x, as given in ref. [4]. After integrating out B, eqn. (2.3) is only an on-shell symmetry, as to be expected. The equation of motion which must be used to maintain this symmetry is obtained from the anti-ghost variation: Dra$~l + ~,p,sD7+~ = 0. Now, without this restriction on $, there are four gauge parameters. However, x only contains three “degrees of freedom”. This suggests that there is further gauge fixing to be done which will reduce the freedom of the ghost field. There must be an additional symmetry in the theory. As we will soon see, this symmetry forces the introduction of two additional commuting fields, 4 and X, and another anti- commuting field, r]. As a result, we will have the full multiplet of fields as given in ref. [4]. We now turn our attention to finding this symmetry while keeping our off-shell BRST invariance. We do not have to look far, as the ghost Lagrangian above has a ghostly symmetry. It is invariant under the following transformations: &aa = i(Da4)a , apa - &B - - ie0[4, xap]” , 4 where 4” is a secondary ghost (or G-ghost) field and 8~ is the BRST operator of the ghost symmetry. To gauge fix this G-symmetry, we begin by writing G?+FP= &~[ico~(Da$~ + sb) + clxaPBaB] , P-5) where & E 81 + &, the c; are arbitrary real constants, X is the G-anti-ghost, s is the G-gauge parameter and b is the G-auxiliary field. However, there are two more symmetries we must worry about. They are the scaling and U symme- tries ‘*I. The global scaling and U symmetry weights of the fields (A, 4, X, ~,LJ,x) are (l,O, 2,1,2) and (0,2, -2,l, -l), respectively. In order to maintain these two invariances (in particular, the latter one), we must take b E eo[qS,q], for some anti-commuting field, 7, which is the transform of X; i.e., &X E 2~. This latter field has scaling and U weights 2 and -1, respectively. Observe that &r] # 0, contrary to what one would expect from traditional BRST gauge fixing. Short of introducing new fields, we find that the ansatz used above is unique. As the algebra of the symmetry, given in eqn. (2.4), c 1oses only up to an ordinary YM gauge transformation [*I, we have 8~7 = -eo$[4, X]. The natural choices for the constants ci are co = -i and cl = 5. This is the case since on manifolds of the form M = Y3 x R1, these choices maintain the time-reversal symmetry of the Lagrangian. Putting all of the above together, we find L: - LO = (f?) + dG))G$‘+Fj= = -ix”‘Da$~p - iqDa$a + iADa-Da4 - i$oX[lliarOa] - ieo4[xaP, xap] + =0(+[rl, rll + $0[A Xl”) + iB”8sap + ~B*‘(Fa~ + Pap) (2.6) = f(F + k)” - ix”‘Dapclp - iqDatia + iXD”Dad - iecJ[tia,$,] - $xd[xaP,xap] + se0W[w] + $0[4, Xl") , 5 where the second equation follows from the first after use of the auxiliary field’s equation of motion. The Lagrangian (2.6) can be brought to the form C - LO = --$l’, where fZ’ is independent of the coupling constant, by a resealing of the fields. It should be observed that for reasons of renormalizability we cannot choose s = 0. Once again naturalness arguments favor the choice s = -$, which maintains the time-reversal symmetry of the Lagrangian. We have thus obtained Witten’s full YM Lagrangian (see eqns. (2.13) and (2.41) in ref. [4]) by straightforward gauge fixing. 3. RENORMALIZATION IN WITTEN’S YM TQFT* Practical quantum field theory computations using the Lagrangian given in eqn. (2.6) become manageable if we make use of the following identification which is suggested by the time-reversal symmetry of the Lagrangian: 1 (3.1) Under time-reversal we have : 4 + A, x -+4, (34 $” -+xa, xa + -fj”. This symmetry suggests a “particle-antiparticle” relationship between 4 and A, and $J” and xa, respectively. This becomes even more suggestive when we rewrite the Lagrangian using the above identification: ,f - lo =i(F + k)2 - ix.J?)F$ + iXD”D& P-3) - * In this section, we will only consider a flat Euclidean background. 6 where @F = I’PD,, with rO = uO ~ uO, rl = o3 @ ia2, r2 = iU2 @CT', IIT3 = O1 @ ia ' (3.4 Here the @ are the standard Pauli matrices. We then have that the I’ll matrices obey the following algebra: {ri,rj} = - 2@, [ri,rj] = +krk, P-5) [r”,ri] = 0. Hence, the I’p satisfy a U(2) algebra.
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